Question

On a snow day, Mason created two snowmen in his backyard. Snowman A was built to a height of 51 inches and Snowman B was built to a height of 29 inches. The next day, the temperature increased and both snowmen began to melt. At sunrise, Snowman A's height decrease by 4 inches per hour and Snowman B's height decreased by 2 inches per hour. Let A(t) represent the height of Snowman A tt hours after sunrise and let B(t)B(t) represent the he ight of Snowman B tt hours after sunrise. Write the equation for each function and determine the interval of time, t,t, when Snowman A is taller than Snowman B.

Answer 1

A ( t ) = -4t + 51

B ( t ) = -2t + 29

t < 11 hours ... [ 0 , 11 ]

Explanation:

Given:-

- The height of snowman A, Ao = 51 in

- The height of snowman B, Bo = 29 in

Solution:-

- The day Mason made two snowmans ( A and B ) with their respective heights ( A(t) and B(t) ) will be considered as the initial value of the following ordinary differential equation.

- To construct two first order Linear ODEs we will consider the rate of change in heights of each snowman from the following day.

- The rate of change of snowman A's height ( A ) is:

`(d h_a)/(dt) = -4`

- The rate of change of snowman B's height ( B ) is:

`(d h_b)/(dt) = -2`

Where,

t: The time in hours from the start of melting process.

- We will separate the variables and integrate both of the ODEs as follows:

`\int {} \, dA= -4 * \int {} \, dt + c\\\\A ( t ) = -4t + c`

`\int {} \, dB= -2 * \int {} \, dt + c\\\\B ( t ) = -2t + c`

- Evaluate the constant of integration ( c ) for each solution to ODE using the initial values given: A ( 0 ) = Ao = 51 in and B ( 0 ) = Bo = 29 in:

`A ( 0 ) = -4(0) + c = 51\\\\c = 51`

`B ( 0 ) = -2(0) + c = 29\\\\c = 29`

- The solution to the differential equations are as follows:

A ( t ) = -4t + 51

B ( t ) = -2t + 29

- To determine the time domain over which the snowman A height A ( t ) is greater than snowman B height B ( t ). We will set up an inequality as follows:

A ( t ) > B ( t )

-4t + 51 > -2t + 29

2t < 22

t < 11 hours

- The time domain over which snowman A' height is greater than snowman B' height is given by the following notation:

Answer: [ 0 , 11 ]